Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16)(17,18)(19,24)(20,28)(21,22)(23,27)(25,31)(26,32)(29,30)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,24)(20,21)(22,28)(23,29)(25,32)(26,31)(27,30)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,2,5,8)(3,6,10,14)(4,7,12,11)(9,13,16,15)(17,20,23,31)(18,25,27,28)(19,26,29,21)(22,30,32,24)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,21)(22,30)(23,31)(24,32)(26,29)(27,28);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,22)(19,20)(23,26)(24,28)(25,30)(27,32)(29,31)(33,37,39,34)(35,40,36,42)(38,41,46,45)(43,48,44,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,22)(18,26)(19,28)(20,30)(21,27)(23,32)(24,31)(25,29)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820457);